Optimal number of measurements for compressed sensing with quadratically decreasing SNR

Abstract

In this paper, we consider a practical signal transmission application with fixed power budget such as radar/sonar. The system is modeled by a linear equation with the assumption that the signal energy per measurement decreases linearly and the noise energy per measurement increases approximately linearly with the increasing of the number of measurements. Thus the SNR decreases quadratically with the number of measurements. This model suggests an optimal operation point different from the common wisdom where more measurements always mean better performance. Our analysis shows that there is an optimal number of measurements, neither too few nor too many, to minimize the mean-squared error of the estimate. The analysis is based on a state evolution technique which is proposed for the approximate message passing algorithm. We consider the Gaussian, Bernoulli-Gaussian and least-favorite distributions in both real and complex domains. Numerical results justify the correctness of our analysis.